On The Variance of Schatten $p$-Norm Estimation with Gaussian Sketching Matrices

TL;DR

This paper analyzes the variance of a Schatten p-norm estimator using Gaussian sketching matrices, providing a new method to compute variance and sharper bounds for the estimator's accuracy.

Contribution

It introduces a procedure to compute the variance of a Schatten p-norm estimator with Gaussian vectors and improves existing bounds on its accuracy.

Findings

Provides a new variance computation method for the estimator.

Offers sharper bounds on the estimator's variance.

Enhances understanding of Gaussian sketching in matrix norm estimation.

Abstract

Monte Carlo matrix trace estimation is a popular randomized technique to estimate the trace of implicitly-defined matrices via averaging quadratic forms across several observations of a random vector. The most common approach to analyze the quality of such estimators is to consider the variance over the total number of observations. In this paper we present a procedure to compute the variance of the estimator proposed by Kong and Valiant [Ann. Statist. 45 (5), pp. 2218 - 2247] for the case of Gaussian random vectors and provide a sharper bound than previously available.